Ben,

I'm going to ammend MX's advice slightly: Lean with the bike. Sometime you can even lean yourself more than the bike, like on flat, loose corners; but you and the bike should be in line with each other. This centers your weight force through the frame. If you lean the bike and sit upright atop it, it will have a tendency to want to spit the tires out from under you.

Check this photo sequence. Elbows out, weighting the bars, body centered in the cockpit, and you can draw a line through bike and torso: